Residential wiring is typically single phase alternating current and employs two active conductors across which a sinusoidal voltage is developed. In contrast, industrial wiring is more commonly for three-phase alternating current and employs three active conductors, each pair of which presents one of three different sinusoidal voltages. Each of these three voltages differs in phase from the others by approximately 120.degree..
The advantages of three-phase power over single phase power are well recognized and include improved generator and motor efficiency, e.g., three-phase power produces less "iron loss" in rotating machinery, and improved power transmission, i.e., with balanced loads the transmission of power in a three-phase system is constant.
Three-phase power is produced by the proper positioning of three windings in an alternator. When one side of each winding is connected at a single point, the windings are said to be connected in the "star" or "Y" configurations. When the windings are connected in ring fashion, with the ends of each winding connected to its neighbors, the windings are said to be connected in the ".DELTA." configuration. Although the following discussion will refer to a Y connected system, the present invention works with both Y and .DELTA. three-phase systems.
Referring now to FIG. 1, a three-phase alternator 10 has three windings connected in a Y configuration, a first end of the windings sharing a common connection O and the second end of the windings connected to exposed terminals A, B and C. Each of the terminals is connected to a conductor 11 for transmitting the generated three-phase power.
Power is obtained from the alternator 10 across pairs of the terminals A, B and C, each pairwise connection being termed a circuit. The, voltage of each circuit is identified by its terminals as: V.sub.AB, V.sub.BC and V.sub.CA respectively, where, for example, V.sub.AB is the voltage at any given time across terminals A and B of the alternator 10.
A phasor diagram 12 of the voltages generated by the alternator 10 consists of three phasors V.sub.AB, V.sub.BC and V.sub.CA spaced equally at 120.degree. from each other about an origin 13. Each phasor has a constant magnitude v.sub.AB, v.sub.BC and v.sub.CA is considered to rotate about the origin 13 at a line frequency f in a phase sequence. The projection of these phasors on an axis 14 traces the voltages wave forms V.sub.AB, V.sub.BC and V.sub.CA with time.
It will be recognized that another similar phasor diagram may be constructed indicating currents flowing between the terminals A, B and C with phasors (not shown) I.sub.A, I.sub.B or I.sub.C having magnitudes i.sub.A, i.sub.B or i.sub.C, respectively. The currents and voltages produced by each circuit of the alternator 10 are referred to generally as "electrical signals".
A three-phase system is "in balance" when the magnitude and angular spacing of the phasors (voltage or current) of each circuit are equal. A balanced system will produce three equal peak amplitude electrical signals, one associated with each circuit, that differ in phase by 120.degree. A balanced three-phase system provides the highest electrical efficiency: three-phase motors, for example, operating on unbalanced three-phase power will experience increased rotor heating which represents a loss of power. Unbalance in a three-phase system may indicate an improper wiring of loads across only one phase or may foreshadow equipment failure, for example: a faulty winding in a motor or alternator or a ground fault condition. For these reasons, determining the state of balance or unbalance of a three-phase system is important.
Balance or lack thereof in a three-phase system may be described by simply listing the phasor magnitudes and angles of the power on the three circuits. This description is not very illuminating however, and therefore it is preferable to describe the balance of a three-phase system through a symmetric component analysis. Such an analysis decomposes the unbalanced system's unsymmetrical phasor diagram into three symmetric phasor components: a positive phase sequence, a negative phase sequence and a zero phase sequence.
Referring to FIG. 2, a phasor diagram of an unbalanced three-phase system 15 may be represented by phasors V'.sub.AB, V'.sub.BC and V'.sub.CA of different magnitudes spaced apart at different angles .alpha., .beta. and .gamma.. As mentioned, the degree of unbalance in such a system is clarified by decomposing the unbalanced system 15 into its symmetric components of: a positive sequence voltage 16, a negative sequence voltage 17 and a zero sequence voltage 18.
The positive sequence voltage 16 is represented by a set of three equally spaced phasors of equal magnitude rotating with frequency f equal to that of the unbalanced system 15. The positive sequence voltage is the balanced part of the unbalanced system which supplies the positive torque to motors and the like connected to the system.
The negative sequence voltage 17 is also a set of three equally spaced phasors of equal magnitude, but rotating in the opposite direction of the positive sequence voltage phasors with a frequency f. The negative sequence voltage represents a counter-rotational torque on motors or power loss caused by unbalance in the unbalanced system 15.
The zero sequence voltage 18 is a non-rotating phasor. The zero sequence voltage 18 represents an unbalanced system 15 such as might be associated with a ground fault in a .DELTA. system or a neutral current in a Y system.
The vector addition of the phasors of the positive sequence voltage 16, the negative sequence voltage 17 and the zero sequence voltage 18 produce the phasors of the unbalanced system 15. The magnitude of the phasors of the positive, negative and zero sequence voltages for a given unbalanced system 15 having phasors V'.sub.AB, V'.sub.BC and V'.sub.CA are calculated as follows: ##EQU1## where, for example, (1.angle.0.degree.) designates a unit length vector at a phase angle of 0.degree. with respect to some fixed reference generally perpendicular to the projection axis 14, and the `*` operator is the scalar or dot product such that: EQU V.sub.1 *V.sub.2 =v.sub.1 v.sub.2 cos.theta. (4)
where .theta. is the included angle between the vectors or phasors. The results of equations (1) to (3) are scalar functions of time. The magnitude of the phasors for each of the symmetrical components 16, 17, and 18 is the peak value of this scalar function during one cycle.
The magnitude of current vectors for the positive sequence current, the negative sequence current, and the zero sequence current may be, likewise, calculated as follows: ##EQU2##
The similarity of the calculations for the sequence voltages given by equations (1)-(3) and the sequence currents given by equations (5)-(7) will be employed to simplify the following discussion in which "positive sequence", for example, refers to either positive sequence current or positive sequence voltage. The sequences of equations (1)-(3) and (5)-(7) will be referred to generally as "symmetric sequences".
Despite the usefulness of symmetric component analyses of unbalanced three-phase systems, the determination of the voltages associated with each component phasor requires complex vector mathematics. Such mathematical analysis may be readily performed on a digital computer, however, the cost of computer hardware capable of performing these calculation in real-time, or near real-time as is ordinarily desired, is prohibitive for many applications in which such information is desired.
For this reason, it is known to perform the vector mathematics needed for symmetrical component analyses with "analog" circuitry in which the vector multiplications are approximated by combinations of phase inversions and phase shifting networks of capacitors and inductors and the scalar additions are accomplished with a summing junction. Such analog systems are capable of near real-time calculation of the symmetrical components of a three-phase system but are extremely frequency sensitive and will provide erroneous decompositions if the frequency of the three-phase power shifts significantly from the frequency at which the networks were calibrated.